I was searching for the Kingsguards vow and the only thing I could find was a collection of quotes here: westeros.org. There they get to the point where it is not clear if the vow transfers automatically to the heir if the previous king dies:

"I swore an oath!"

"To Joffrey, not to Tommen."

"Aye..."

taken from the linked site

So we can assume that IF they don't have to serve the new king automatically, they can decide to leave the Kingsguard once the king they vowed to dies. So that could potentially be one point to get rid of the white cloak.
I have not read the books yet, but in the tv series:

When Tyrion is in trial for killing Joffrey, Jaime offers his father to leave the Kingsguard so he can take his rightful place as the ruler of Casterly Rock if he spares Tyrions life.

So: are Kingsguards allowed to leave their position at any time or can they only get out by getting dumped? Is this an exception for the Lord Commander? Or (if the assumption mentioned above is right) do they have to wait for their king to die?

I am asking, because there is so much said about the vow of the Night's Watch in the tv series, but they never say anything about the Kingsguards, at least I do not remember.

I'm fairly sure it's a vow for life, very similar to the one taken by the Black Watch. Remember Ser Barristan's reaction when he was retired! I don't have any sources to hand though.
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LiathAug 8 '14 at 8:37

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Had to be said: One does not simply walk out of Kingsguard.
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DVKAug 8 '14 at 13:55

2 Answers
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they are sworn to protect their king and the royal family with their
own lives, to obey his commands, and to keep his secrets. They are
sworn for life and are forbidden from owning land, taking a wife, or
fathering children.

-- A Game of Thrones, Chapter 6, Catelyn.

The king (pretty much does whatever he wants, so he) can dismiss someone off the guard, like Joffrey did with Selmy; or like Tywin wanted to do with Jaime, when the former was king Regent.

But a person cannot leave on his own. That is why they were after Sandor after he left the Kingsguard.

The dismissing of Selmy was unprecedented (at least in the books). In aSoS2, Tywin says to Jaime that Cersei was a fool to let Selmy go, but now that door has been opened, Kingsguard can be dismissed again. So it sounds like Joffrey is more of an 'I do whatever I want' sort of king than any of his predecessors ... not surprising really.
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rand al'thorAug 8 '14 at 10:23

When the Kingsguard was founded by the Targaryan dynasty, the vows they took were very much like those of the Night's Watch:

They served until death - technically even a sick or permanently disabled member was still a Kingsguard as long as he was alive.

They could not hold lands, though they could apparently hold certain titles (Jaime was named Warden of the East after Jon Arryn died and I believe several Kingsguard were also Hand of the King)

They could neither marry not father children.

When Robert took the throne from Aerys, in theory all (two - Jaime and Barristan Selmy) of the surviving Kingsguard would have been considered traitors to the current crown and punished, but Robert pardoned them both and they kept their positions. Thus, the Baratheon Kingsguard appears to be the same order as the previous one, so they would follow the same vows.

However, the King can dismiss a Kingsguard at any time, since they serve the king at his pleasure. Had Jaime's deal with his father gone as planned, it would have been easy enough to have Tommen dismiss him at the next possible opportunity.

'considered traitors to the current crown and punished' - that only goes for Selmy. Jaime (Kingslayer) was a traitor to the previous crown, which is just fine!
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rand al'thorAug 8 '14 at 10:26

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he was still officially pardoned by Robert before being allowed back into the Kingsguard, since he WAS a murderer...
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Michael EdenfieldAug 8 '14 at 12:18

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@randal'thor: I don't think the fact that Jaime betrayed Aerys necessarily would mean he hadn't already "betrayed" Robert in the same way Barristan did, that is to say opposing his rebellion. Basically, it was victor's justice for Robert, he could declare pretty much anything as long as he maintained a workable coalition of everyone left alive. So Robert got to pick both who he pardoned, and the details of what he pardoned them for :-)
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Steve JessopAug 8 '14 at 12:36